Method and apparatus for soft detection of high order qam symbols in mimo channels

ABSTRACT

Methods and apparatus for soft MIMO detection of high order QAM with initial candidate reduction are described. A method includes receiving a plurality of signals including Q-order QAM symbols; determining a reduced candidate set including C potential candidates, where C is less than Q; calculating Euclidean distances (EDs) based on the reduced candidate set; and generating LLR information based on the calculated EDs.

PRIORITY

The present application claims priority under 35 U.S.C. §119(e) to U.S.Provisional Application Nos. 62/152,366, and 62/250,268, which werefiled in the U.S. Patent and Trademark Office on Apr. 24, 2015 and Nov.3, 2015, respectively, the content of each of which is incorporatedherein by reference.

FIELD OF THE DISCLOSURE

The present disclosure relates generally to soft detection of high orderquadrature amplitude modulation (QAM) symbols, and more specifically, tosoft detection of high order QAM symbols in multiple input multipleoutput (MIMO) channels, with or without prior information.

BACKGROUND

Increasing demand for data transmissions in wireless networks hasincreased the need for higher throughput systems. High order modulationand/or MIMO set-up can address the demand for high throughput. Forexample, 256-QAM signaling has been adopted by the 3^(Rd) GenerationPartnership Project (3GPP) group in Long Term Evolution (LTE)-Release 12to increase LTE system throughput. Further, the currently developingInstitute of Electrical and Electronics Engineers (IEEE) 802.11axstandard is considering 1024-QAM to further increase Wi-Fi throughput.

However, hardware implementation complexity of maximum likelihood (ML)detection of MIMO channels increases exponentially with the number oftransmitted layers and modulation order, which makes real-world hardwareimplementation infeasible. For example, the hardware complexity for softdetection of 256-QAM MIMO with two transmitted layers is roughly 16times the hardware complexity of 64-QAM MIMO with two transmittedlayers. Therefore, the use of sub-optimal schemes is inevitable inpractical hardware implementation.

Some sub-optimal detection schemes have already been introduced. Themost common scheme which reduces optimal ML complexity is obtained withmax-log-MAP (MLM) approximation. However, high order modulationsignaling such as 256-QAM still makes hardware implementation of MLMscheme infeasible.

Soft list sphere decoding (LSD) has also been introduced as analternative to further reduce complexity in soft detection of coded MIMOchannels. However, the variable complexity of LSD, as well as thecomplexity of search space selection of LSD, introduce new challenges inhardware implementation.

SUMMARY

In accordance with an aspect of the present disclosure, a method isprovided for log-likelihood ratio (LLR) generation for soft detection ofQAM symbols in MIMO coded channels. The method includes receiving aplurality of signals including Q-order QAM symbols; determining areduced candidate set including C potential candidates, where C is lessthan Q; calculating Euclidean distances (EDs) based on the reducedcandidate set; and generating LLR information based on the calculatedEDs.

In accordance with another aspect of the present disclosure, anapparatus is provided for soft detection of QAM symbols in MIMO codedchannels. The apparatus includes a plurality of antennas; and a MIMOdetector that receives, via the plurality of antennas, a plurality ofsignals including Q-order QAM symbols, determines a reduced candidateset including C potential candidates, where C is less than Q, calculatesEDs based on the reduced candidate set, and generates LLR informationbased on the calculated EDs.

In accordance with another aspect of the present disclosure, a system onchip is provided, which includes a MIMO detector that receives aplurality of signals including Q-order QAM symbols, determines a reducedcandidate set including C potential candidates, where C is less than Q,calculates EDs based on the reduced candidate set, and generates LLRinformation based on the calculated EDs; and a decoder that decodes thesignals using the LLR information.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other aspects, features, and advantages of certainembodiments of the present disclosure will be more apparent from thefollowing detailed description taken in conjunction with theaccompanying drawings, in which:

FIG. 1 is a flowchart illustrating a method of generating LLR for softdetection of MIMO coded channels according to an embodiment of thepresent disclosure;

FIG. 2 illustrates a reduced initial candidate set according to anembodiment of the present disclosure;

FIGS. 3 and 4 are graphs illustrating block error rate (BLER)performances of an initial candidate reduction (ICR) scheme and MLM forlow and high antenna correlation cases, respectively, and different coderates (CRs), according to an embodiment of the present disclosure;

FIGS. 5 and 6 are graphs illustrating BLER performances of an ICR schemeusing minimum mean square error (MMSE) with and without cross-priorinformation and MLM for low and high antenna correlation cases,respectively, and different CRs, according to an embodiment of thepresent disclosure; and

FIG. 7 is a block diagram illustrating a receiving apparatus accordingto an embodiment of the present disclosure.

DETAILED DESCRIPTION

Various embodiments of the present disclosure will now be described indetail with reference to the accompanying drawings. In the followingdescription, specific details such as detailed configuration andcomponents are merely provided to assist the overall understanding ofthese embodiments of the present disclosure. Therefore, it should beapparent to those skilled in the art that various changes andmodifications of the embodiments described herein can be made withoutdeparting from the scope and spirit of the present disclosure. Inaddition, descriptions of well-known functions and constructions areomitted for clarity and conciseness.

Various embodiments may include one or more elements. An element mayinclude any structure arranged to perform certain operations. Althoughan embodiment may be described with a limited number of elements in acertain arrangement by way of example, the embodiment may include moreor less elements in alternate arrangements as desired for a givenimplementation. It is worthy to note that any reference to “oneembodiment” or “an embodiment” means that a particular feature,structure, or characteristic described in connection with the embodimentis included in at least one embodiment. The appearances of the phrase“in one embodiment” in various places in this specification are notnecessarily all referring to the same embodiment.

The present disclosure has been made to address at least the problemsand/or disadvantages described above and to provide at least theadvantages described below.

An aspect of the present disclosure is to provide low complexity schemesfor soft detection of high order QAM symbols in MIMO channels, with orwithout prior information.

Another aspect of the present disclosure is to provide an ICR scheme toreduce the number of ED calculations during LLR generation.

Another aspect of the present disclosure is to provide a scheme forinitial candidate set selection relying on linear MMSE detection.

Another aspect of the present disclosure is to provide a scheme forinitial candidate set selection by detecting only I and Q signs of atarget layer, which simplifies MMSE detection.

Another aspect of the present disclosure is to improve accuracy of aninitial candidate set, by using prior information in initial candidateset selection with MMSE soft interference cancellation (MMSE-SIC).

Another aspect of the present disclosure is to improve accuracy of aninitial candidate set, by providing an approximation over MMSE-SIC byusing self-prior information to avoid computational complexity.

Herein, the terminology “search space” and “initial candidate set” maybe used interchangeably.

The present disclosure first describes 256 QAM signaling detection inMIMO channels with two transmitted layers with the understanding thatthe schemes of the present disclosure can be generalized to other QAMmodulation orders and higher rank MIMO channels. To reduce thecomplexity of the search space (or equivalently candidate set)selection, a fixed complexity search space is provided with 128 initialcandidates, such that in the selection of the search space (or theinitial candidate set) only an initial estimate of an in-phase (I) signand a quadrature (Q) sign of the transmitted signal is obtained. Thepresent disclosure then describes the use of a priori information (fromdecoder output) to improve initial candidate set selection accuracy.MMSE-SIC is described in use of both cross-prior and self-priorinformation. Thereafter, a low complexity scheme is provided usingself-prior information, wherein an initial MMSE estimation of thetransmitted signal is obtained without any prior information, then priorinformation on I and Q signs of the transmitted signal is applied (fromdecoder output either in re-transmission or in iterative detection anddecoding (IDD)) in hard detection of (slicing over) the initial MMSEestimated value. This scheme provides slicer boundaries as a linearfunction of the self-prior information, which decreases complexitysignificantly.

Although descriptions of various embodiments of the present disclosurewill be provided below, which focus on 256-QAM signaling detection inMIMO channels with two transmitted layers, it will be appreciated by aperson having ordinary skill in the art that the present disclosure isalso applicable to other QAM modulation orders, e.g., 512-QAM and1024-QAM, and higher rank MIMO channels.

Additionally, to reduce the complexity of the search space (or candidateset) selection, in accordance with an embodiment of the presentdisclosure, a reduced size, fixed complexity search space with 128initial candidates is utilized (instead of 256 initial candidates for256-QAM signaling detection), such that in the selection of an initialcandidate set only an initial estimate of the I and Q sign of thetransmitted signal is required. However, it will be appreciated by aperson having ordinary skill in the art that the present disclosure isalso applicable to other reduced size initial candidate sets, e.g., 64or 32.

In accordance with another embodiment of the present disclosure, apriori information (e.g., output from a decoder) is used to improveinitial candidate set selection accuracy.

ICR Scheme

FIG. 1 is a flowchart illustrating a method of generating LLR for softdetection of MIMO coded channels according to an embodiment of thepresent application.

Referring to FIG. 1, a receiving apparatus including a plurality ofantennas receives a plurality of signal streams from a plurality oftransmitted signals at 105.

In a point-to-point MIMO system with 2 transmit antennas and n_(R)receiver antennas, a channel model for MIMO rank 2 can be written asshown in Equation (1) below.

y=Hx+n=h ₀ x ₀ +h ₁ x ₁ +n   (1)

In Equation (1), y=[y₀, . . . , y_(r−1)]^(T) is an n_(R)×1 receivesignal vector, x=[x₀, x₁]^(T) is a 2×1 transmit signal vector, H=[h₀,h₁] is an n_(R)×2 channel coefficient matrix, h_(i)=[h_(i,0), . . . ,h_(i,n) _(R) ⁻¹]^(T), h_(i,j) represents a channel between i-th transmitand j-th receive antennas, and n is an additive white Gaussian noisevector with covariance E{nn^(H)}=σ²I. In addition, each symbol x_(i)carries a bit vector b_(i)=(b_(i,0) . . . b_(i,M−1))∈ {0,1}^(M).Therefore, an i-th layer's transmit symbol x_(i) is chosen from 256-QAMconstellation with 256 constellation points and is normalized such thatE{|x_(i)|²}=1.

For the channel model shown in Equation (1), a soft maximum likelihood(ML) receiver to generate a posteriori log-likelihood ratio (LLR) for1-th bit b_(0,l) is shown in Equation (2).

$\begin{matrix}{\begin{matrix}{{L_{A}\left( b_{0,l} \right)} = {{\log \frac{P\left( {b_{0,l} = {0y}} \right)}{P\left( {b_{0,l} = {1y}} \right)}} = {\log \frac{{P\left( {{yb_{0,l}} = 0} \right)}{P\left( {b_{0,l} = 0} \right)}}{{P\left( {{yb_{0,l}} = 1} \right)}{P\left( {b_{0,l} = 1} \right)}}}}} \\{= {{\log \frac{\sum\limits_{{x_{0}\text{:}b_{0,l}} = 0}{\sum\limits_{x_{1}}{{P\left( {{yx_{0}},x_{1}} \right)}{\prod\limits_{{({m,n})} \neq {({i,l})}}^{\;}\; {P\left( b_{m,n} \right)}}}}}{\sum\limits_{{x_{0}\text{:}\mspace{14mu} b_{0,l}} = 1}{\sum\limits_{x_{1}}{{P\left( {{yx_{0}},x_{1}} \right)}{\prod\limits_{{({m,n})} \neq {({i,l})}}^{\;}\; {P\left( b_{m,n} \right)}}}}}} + {L_{a}\left( b_{0,l} \right)}}} \\{= {{\log \frac{\sum\limits_{{x_{0}\text{:}\mspace{14mu} b_{0,l}} = 0}{\sum\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}^{\;}\; {P\left( b_{m,n} \right)}}}}}{\sum\limits_{{x_{0}\text{:}\mspace{14mu} b_{0,l}} = 1}{\sum\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}^{\;}\; {P\left( b_{m,n} \right)}}}}}} + {L_{a}\left( b_{0,l} \right)}}}\end{matrix}\quad} & (2)\end{matrix}$

In Equation (2),

${p\left( {yx} \right)} \sim {\exp \left( {{- \frac{1}{\sigma^{2}}}{{y - {Hx}}}^{2}} \right)}$and${L_{a}\left( b_{0,l} \right)} = \frac{P\left( {b_{0,l} = 0} \right)}{P\left( {b_{i} = 1} \right)}$

is a priori LLR for b_(0,l). However, the direct implementation of theLLR calculation above involves searching over 256×256 EDs, imposing aserious burden for hardware implementation. To reduce the number of EDsinvolved in LLR calculations and avoid exponential operations, LLRgeneration for x₀=I+jQ is considered, where an MLM approximation methodfor soft non-linear joint MIMO detection replaces the summationoperations in Equation (2) with max operations, as shown in Equation(3).

$\begin{matrix}{\begin{matrix}{{L_{A}\left( b_{0,l} \right)} = {{\log \frac{\max\limits_{{x_{0}\text{:}b_{0,l}} = 0}{\max\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}^{\;}\; {P\left( b_{m,n} \right)}}}}}{\max\limits_{{x_{0}\text{:}\mspace{14mu} b_{0,l}} = 1}{\max\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}^{\;}\; {P\left( b_{m,n} \right)}}}}}} + {L_{a}\left( b_{0,l} \right)}}} \\{= {{\frac{1}{\sigma^{2}}\begin{pmatrix}{{\min\limits_{b_{0,l} = 1}\left( {{{y - {Hx}}}^{2} - {\frac{1}{2}{\sum\limits_{{({m,n})} \neq {({i,l})}}^{\;}{\left( {- 1} \right)^{b_{m,n}}{L_{a}\left( b_{m,n} \right)}}}}} \right)} -} \\{\min\limits_{b_{0,l} = 0}\left( {{{y - {Hx}}}^{2} - {\frac{1}{2}{\sum\limits_{{({m,n})} \neq {({i,l})}}^{\;}{\left( {- 1} \right)^{b_{m,n}}{L_{a}\left( b_{m,n} \right)}}}}} \right)}\end{pmatrix}} +}} \\{{L_{a}\left( b_{0,l} \right)}}\end{matrix}\quad} & (3)\end{matrix}$

As shown above for Equation (3), even for the MLM scheme, 256 EDs in LLRgeneration of b_(0,l) are required to be calculated.

Referring again to FIG. 1, at 110, the receiving apparatus, e.g., a MIMOdetector therein, determines a reduced initial candidate set. Inaccordance with an embodiment of the present disclosure, an ICR schemeis provided with an initial candidate set of 128, in order to reduce thenumber of ED calculations.

Similar to sphere decoding (SD), the search space (or candidate set) ofMIMO detection may be limited to lattice points around the Babai point.In accordance with an embodiment of the present disclosure, the size ofthe search space is reduced in order to reduce the computationalcomplexity with the selection of search space around an initialestimation of x₀. For example, the MIMO detection search space islimited to 128 points around an initial estimate of x₀.

In accordance with an embodiment of the present disclosure, a linearMMSE estimation of x₀ may be used for selection of a search space (or aninitial candidate set). The MMSE estimation {circumflex over (x)}₀ maybe obtained using Equation (4).

$\begin{matrix}{\begin{matrix}{{\hat{x}}_{0} = {\frac{1}{{ch}_{pow}}\left( {{\left( {{h_{1}}^{2} + \sigma^{2}} \right)h_{0}^{H}y} - {h_{0}^{H}h_{1}h_{1}^{H}y}} \right)}} \\{= \frac{{\left( {{\left( {{h_{1}}^{2} + \sigma^{2}} \right){h_{0}}^{2}} - {{h_{0}^{H}h_{1}}}^{2}} \right)x_{0}} + {\sigma^{2}{{h_{0}^{H}h_{1}}}^{2}x_{1}} + n^{\prime}}{{ch}_{pow}}}\end{matrix}\quad} & (4)\end{matrix}$

In Equation (4), ch_(pow)=(|h₁|²+σ²)(|h₀|²+σ²)−|h₀ ^(H)h₁|² andn′=(|h₁|²+σ²)h₀ ^(H) n−h₀ ^(H)h₁h₁ ^(H)n.

For soft decoding, in order to include all possibilities of the bits inthe selected candidate set, i.e., for each i ∈ {0, . . . , 7}, at leastone symbol with b_(0,i)=0 and one with b_(0,i)=1 is used. There may bemany possibilities for the initial candidate set to satisfy thiscondition for a given number of candidates. However, for 128 initialcandidates, in accordance with an embodiment of the present disclosure,an initial candidate set is determined as illustrated in FIG. 2, forI=Re{{circumflex over (x)}₀}>0 and Q=Im{{circumflex over (x)}₀}>0. Theinitial candidate set illustrated in FIG. 2 ensures that there is atleast one candidate for each bit value at each bit position.

Referring to FIG. 2, an initial candidate set is illustrated, where theinitial estimation is located inside box 200. For other quadrants, asymmetric point may be selected.

In accordance with an embodiment of the present disclosure, the initialcandidate set selection only requires detection of the I and Q signs ofx₀, which significantly simplifies linear MMSE detection. Instead ofcalculating {circumflex over (x)}₀, any scaled version of I and Q may beused in candidate set selection. Therefore, instead of {circumflex over(x)}₀, ch_(pow){circumflex over (x)}₀ may be used for the initialcandidate set selection and only the I and Q signs of (|h₁|²+σ²)h₀^(H)y−h₀ ^(H)h₁h₁ ^(H)y are determined. As a result, the initialestimation complexity is significantly reduced in comparison withoriginal MMSE estimation, as there is no need for ch_(pow) calculationand division by ch_(pow) as shown in Equation (4) above.

Referring again to FIG. 1, after the reduced initial candidate set isdetermined, EDs are calculated at 115 and an LLR for x₀ is generated at120.

Accordingly, if X₁₂₈({circumflex over (x)}₀) is defined as the set of128 initial candidates selected by the initial MMSE estimation{circumflex over (x)}₀ (e.g., the constellation points inside the shadedregions in FIG. 2 represent X₁₂₈({circumflex over (x)}₀) for I>0 andQ>0), then a posteriori LLR of b_(0,l), i.e., L_(A)(b_(0,l)), may begenerated using Equation (5) below.

$\begin{matrix}{{L_{A}\left( b_{0,l} \right)} \approx {{\log \frac{\max\limits_{{{{x_{0} \in {X_{128}{({\hat{x}}_{0})}}}\&}b_{0,l}} = 0}{\max\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}^{\;}\; {P\left( b_{m,n} \right)}}}}}{\max\limits_{{{{x_{0} \in {X_{128}{({\hat{x}}_{0})}}}\&}b_{0,l}} = 1}{\max\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}^{\;}\; {P\left( b_{m,n} \right)}}}}}} + {L_{a}\left( b_{0,l} \right)}}} & (5)\end{matrix}$

By applying Max-Log approximation and further by limiting the searchspace to a subset of constellation points, a non-linear estimation ofthe actual LLR is obtained. Therefore, LLR clipping may be applied tolimit the amount of error in LLR generation using the ICR scheme incomparison with a conventional LLR calculation as shown above inEquation (2).

At 125, the generated LLR for x₀ is provided to a decoder of thereceiving apparatus.

Performance of the ICR scheme, as described above, under realistic LTEchannel conditions may be tested using simulation parameters as shown inTable 1 below.

TABLE 1 Simulation Parameters System bandwidth 10 MHz FFT Size 1024  Number of transmit 2 antennas Number of receive 2 antennas Antennacorrelation Low, medium, and high

FIGS. 3 and 4 are graphs illustrating BLER performances of the ICRscheme and MLM for low and high antenna correlation cases and differentCRs according to an embodiment of the present disclosure. Specifically,FIG. 3 illustrates a comparison between BLER performances of an ICRscheme according to an embodiment of the present disclosure and a fullycalculated MLM detection for CRs of 0.6 and 0.8, with low antennacorrelation, and FIG. 4 illustrates a comparison between BLERperformances of an ICR scheme according to an embodiment of the presentdisclosure and a fully calculated MLM detection for a CR of 0.6, withhigh antenna correlation.

As illustrated in FIGS. 3 and 4, the BLER performance of a fullycalculated MLM detection, e.g., as shown in Equation (3) above, may becompared with the BLER performance of the ICR scheme, e.g., as shown inEquation (5) above. Notably, for the low antenna correlation, e.g.,antenna correlation as defined in the 3GPP standard, any gap between MLMand the ICR scheme is negligible. That is, as illustrated in FIG. 3, theMLM plots for both rates substantially overlap the ICR scheme plots.

However, as illustrated in the high antenna correlation case of FIG. 4,there is more than 0.2 dB gap. Higher antenna correlation results inhigher interference level in initial MMSE detection, where for signal tointerference-plus-noise ratio (SINR) of {circumflex over (x)}₀ inEquation (4):

$\begin{matrix}{{{SIN}\; {R\left( {\hat{x}}_{0} \right)}} = {\frac{{h_{0}}^{2}}{\sigma^{2}} - \frac{{{h_{0}^{H}h_{1}}}^{2}}{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}}} & (6)\end{matrix}$

As can been observed in Equation (6), for fixed σ², |h₀|² and |h₁|²,SINR is a decreasing function of |h₀ ^(H)h₁|². As a result, MMSEdetection quality is a decreasing function of antenna correlation |h₀^(H)h₁|².

Based on this observation, in accordance with an embodiment of thepresent disclosure, prior information may be used in an initial MMSEestimation using the ICR scheme to improve the quality of the estimationand the overall soft detection performance.

MMSE-SIC with Prior Information

Herein, x₀ may be referred to as a self layer, i.e., the desired layerin soft detection, and x₁ may be referred to as a cross layer. In oneembodiment of the present disclosure, an MMSE-SIC method utilizes priorinformation on the self layer (hereinafter, referred to as “self-priorinformation”) and prior information on the cross layer (hereinafter,referred to as “cross-prior information”) in MMSE detection of x₀. Inanother embodiment, self prior information is used to avoid thecomputation complexity of the MMSE-SIC method.

In accordance with an embodiment of the present disclosure, self-priorinformation on x₀ and/or cross-prior information on x₁ may be used in anMMSE-SIC scheme to improve the initial candidate selection quality, asshown in Equation (7).

$\begin{matrix}{{\hat{x}}_{0} = {{{\frac{1}{{ch}_{pow}}\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}h_{0}^{H}y} - {\lambda_{0}^{2}\lambda_{1}^{2}h_{0}^{H}h_{1}h_{1}^{H}y} - {\sigma^{2}\lambda_{0}^{2}h_{0}^{H}h_{1}\mu_{1}} - {\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}{h_{0}}^{2}} - {\lambda_{0}^{2}\lambda_{1}^{2}{{h_{0}^{H}h_{1}}}^{2}}} \right)\mu_{0}}} \right)} + \mu_{0}} = {\frac{1}{{ch}_{pow}}\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}h_{0}^{H}y} - {\lambda_{0}^{2}\lambda_{1}^{2}h_{0}^{H}h_{1}h_{1}^{H}y} - {\sigma^{2}\lambda_{0}^{2}h_{0}^{H}h_{1}\mu_{1}} + {{\sigma^{2}\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)}\mu_{0}}} \right)}}} & (7)\end{matrix}$

In Equation (7), ch_(pow)=(|h₁|²+σ²)(|h₀|²+σ²)−|h₀ ^(H)h₁|²,μ_(i)=E{x_(i)}, and

${E\left\{ {\begin{bmatrix}{x_{0} - \mu_{0}} \\{x_{1} - \mu_{1}}\end{bmatrix}\begin{bmatrix}{x_{0}^{*} - \mu_{0}^{*}} & {x_{1}^{*} - \mu_{1}^{*}}\end{bmatrix}} \right\}} = {\begin{bmatrix}\lambda_{0}^{2} & 0 \\0 & \lambda_{1}^{2}\end{bmatrix}.}$

In accordance with an embodiment of the present disclosure, a priori LLRfrom decoder output, e.g., in IDD, and/or from a previously receivedsignal, e.g., from a hybrid automatic repeat request (HARQ) buffer inre-transmission, may be used to calculate μ_(i) and λ_(i) ².

However, an exact calculation of μ_(i) and λ_(i) ² involves exponentialterms, which adds computational complexity.

Therefore, in accordance with an embodiment of the present disclosure,an approximation over MMSE-SIC is provided, which uses self priorinformation to reduce computational complexity often associated withμ_(i) and λ_(i) ² calculation.

Approximation Over MMSE-SIC with Prior Information

Without loss of generality, it is assumed in the description below thatthere is no cross-prior information available, i.e., μ₁=0 and λ₁ ²=1.Therefore, with use of self-prior information in the initial MMSEdetection, s₀(μ₀, λ₀) may be obtained and then used for quadrature phaseshift keying (QPSK) slicing (I and Q sign detection) without any priorinformation, as shown in Equation (8).

$\begin{matrix}{\begin{matrix}{{s_{0}\left( {\mu_{0},\lambda_{0}} \right)} = {{\left( {{h_{1}}^{2} + \sigma^{2}} \right)\lambda_{0}^{2}h_{0}^{H}y} - {\lambda_{0}^{2}h_{0}^{H}h_{1}h_{1}^{H}y} + {{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}\mu_{0}}}} \\{= {{\lambda_{0}^{2}{s_{0}\left( {0,1} \right)}} + {{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}\mu_{0}}}}\end{matrix}\quad} & (8)\end{matrix}$

Without loss of generality, the sign detection of I is equivalent to thedetection of b_(0,0) from s₀(μ₀, λ₀). Therefore, with QPSK slicingwithout prior information over s₀(μ₀, λ₀), b_(0,0) may be determined asshown in Equation (9).

$\begin{matrix}{b_{0,0} = \left\{ \begin{matrix}0 & {{{RE}\left\{ {s_{0}\left( {\mu_{0},\lambda_{0}} \right)} \right\}} > 0} \\1 & {{{RE}\left\{ {s_{0}\left( {\mu_{0},\lambda_{0}} \right)} \right\}} \leq 0}\end{matrix} \right.} & (9)\end{matrix}$

The above operation is equivalent with slicing over s₀(0,1)=(|h₁|²+σ²)h₀^(H)y−h₀ ^(H)h₁h₁ ^(H)y by changing slicing boundaries as shown inEquation (10).

$\begin{matrix}{b_{0,0} = \left\{ \begin{matrix}0 & {{{RE}\left\{ {s_{0}\left( {0,1} \right)} \right\}} > {- \frac{{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}{RE}\left\{ \mu_{0} \right\}}{\lambda_{0}^{2}}}} \\1 & {{{RE}\left\{ {s_{0}\left( {0,1} \right)} \right\}} \leq {- \frac{{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}{RE}\left\{ \mu_{0} \right\}}{\lambda_{0}^{2}}}}\end{matrix} \right.} & (10)\end{matrix}$

On one hand, in calculating Re{μ₀}, only even positioned bits, i.e.,b_(0,i) for i ∈ {0,2,4,6}, are effective. On the other hand, the priorinformation on b_(0,0) determines the sign of Re{μ₀}. It is expectedthat the most significant effect in detection with prior of b_(0,0)comes from prior information on b_(0,0) and the effect of priorinformation from other bits to be averaged out. Therefore, to reduce thecomplexity of the slicing, only the prior information on b_(0,0) isconsidered in calculation of μ₀ and λ₀ ² and the other available priorinformation is ignored, i.e., assuming

${P\left( {b_{0,i} = 0} \right)} = {{P\left( {b_{0,i} = 1} \right)} = \frac{1}{2}}$

for i≠0.

Hence, with only use of prior information on b_(0,0), for 256-QAM,

${{RE}\left\{ \mu_{0} \right\}} = {{{\frac{8}{\sqrt{170}}{P\left( {b_{0} = 0} \right)}} - {\frac{8}{\sqrt{170}}{P\left( {b_{0} = 1} \right)}}} = {\frac{8}{\sqrt{170}}\left( {{2{P\left( {b_{0} = 0} \right)}} - 1} \right)}}$

and λ₀ ²=E{|x₀−μ₀|²}=E{RE{x₀−μ₀}²+Im{x₀−μ₀}²}=1−RE{μ₀}², whereE{|x₀|²}=1.

Defining

${TH}_{0} = \frac{{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}{RE}\left\{ \mu_{0} \right\}}{\lambda_{0}^{2}}$

and L=L_(a)(b_(0,0)) and using

${{P\left( {b_{0} = 0} \right)} = \frac{^{L_{a}{(b_{0})}}}{1 + ^{L_{a}{(b_{0})}}}},$

TH₀ may be determined as shown in Equation (11).

$\begin{matrix}\begin{matrix}{{TH}_{0} = {\frac{8{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}}{\sqrt{170}}*\frac{{2{P\left( {b_{0} = 0} \right)}} - 1}{1 - {\frac{64}{170}*\left( {{2{P\left( {b_{0} = 0} \right)}} - 1} \right)^{2}}}}} \\{= {\frac{8{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}}{\sqrt{170}}*\frac{\frac{^{L} - 1}{^{L} + 1}}{1 - {\frac{64}{170}\frac{\left( {^{L} - 1} \right)^{2}}{\left( {^{L} + 1} \right)^{2}}}}}} \\{= {8\sqrt{170}{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}\frac{\left( {^{L} - 1} \right)\left( {^{L} + 1} \right)}{{170\left( {^{L} + 1} \right)^{2}} - {64\left( {^{L} - 1} \right)^{2}}}}} \\{= {8\sqrt{170}{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}\frac{^{2L} - 1}{{106^{2L}} + {468^{L}} + 106}}}\end{matrix} & (11)\end{matrix}$

Using Taylor expansion of

$\frac{^{2L} - 1}{{106^{2L}} + {468^{L}} + 106}$

at L=0 as

${\frac{^{2L} - 1}{{106^{2L}} + {468^{L}} + 106} = {{\frac{L}{340} + \frac{11L^{3}}{346800} - \frac{1099\; L^{5}}{58956000} + {O\left( L^{7} \right)}} \approx \frac{L}{340}}},{TH}_{0}$

may be approximated as shown in Equation (12).

$\begin{matrix}{{{TH}_{0} \approx {8\sqrt{170}{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}*\frac{L}{340}}} = {\frac{4{\sigma^{2}\left(  \right.}h_{1}\left. ^{2}{+ \sigma^{2}} \right)}{\sqrt{170}}L}} & (12)\end{matrix}$

Therefore, based on the foregoing, a decision rule may be obtained asshown in Equation (13).

$\begin{matrix}{b_{0} = \left\{ \begin{matrix}0 & {{{RE}\left\{ {s_{0}\left( {0,1} \right)} \right\}} > {{- \frac{4{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}}{\sqrt{170}}}{L_{a}\left( b_{0,0} \right)}}} \\1 & {{{RE}\left\{ {s_{0}\left( {0,1} \right)} \right\}} \leq {{- \frac{4{\sigma^{2}\left( {{h_{1}}^{2} + \sigma^{2}} \right)}}{\sqrt{170}}}{L_{a}\left( b_{0,0} \right)}}}\end{matrix} \right.} & (13)\end{matrix}$

Accordingly, the threshold in sign detection of I is a linear functionof the prior information, which reduces hardware implementationcomplexity.

FIGS. 5 and 6 are graphs illustrating BLER performances of an ICR schemeusing MMSE with and without cross-prior information and MLM for low andhigh antenna correlation cases, respectively, and CRs, according to anembodiment of the present disclosure. Specifically, FIG. 5 illustrates acomparison among BLER performances of an ICR scheme using MMSE withcross-prior information, an ICR scheme using MMSE without cross-priorinformation, and a fully calculated MLM detection for CRs of 0.6 and0.8, respectively, with low antenna correlation, and FIG. 6 illustratesa comparison among BLER performances of an ICR scheme using MMSE withcross-prior information, an ICR scheme using MMSE without cross-priorinformation, and a fully calculated MLM detection for a CR of 0.6, withhigh antenna correlation.

As illustrated in FIG. 5, for low antenna correlation, the performancedifferences between the ICR scheme using MMSE with cross-priorinformation, the ICR scheme using MMSE without cross-prior information,and the fully calculated MLM detection are negligible. That is, thedifferences between the performance differences between the ICR schemeusing MMSE with cross-prior information, the ICR scheme using MMSEwithout cross-prior information, and the fully calculated MLM detectionare so small that the plots thereof appear as single line.

Further, as illustrated in FIG. 6, performance gain in using cross priorinformation increases as antenna correlation increases, where the use ofcross prior information reduces the interference from the cross layer inthe initial MMSE detection. Further, for high antenna correlation, theICR scheme using MMSE with cross-prior information improves theperformance by about 0.1 dB as compared to the ICR scheme using MMSEwithout cross-prior information. Additionally, for high antennacorrelation, the ICR scheme using MMSE without cross-prior informationhas better performance than MLM by about 0.1 dB gain.

FIG. 7 is a block diagram illustrating a receiving apparatus accordingto an embodiment of the present application.

Referring to FIG. 7, the receiving apparatus, e.g., a user equipment(UE), includes a plurality of antennas 705, a MIMO detector 710, and adecoder 715. While the receiving apparatus may include additionalcomponents, e.g., a demodulator, an interleaver, a deinterleaver, etc.,as these additional components are not directly related to the presentdisclosure, their illustration and description has been omitted herein.Further, while the MIMO detector 710 and the decoder 715 are illustratedas separate components in FIG. 7, these components could also becombined in to a single processing unit, such as a modem chipset.

The receiving apparatus illustrated in FIG. 7 may perform the method asillustrated in FIG. 1, including the multiple variations of the ICRscheme, the MMSE-SIC scheme using the prior information, and theapproximation over MMSE-SIC scheme using the prior information, asdescribed above.

Specifically, the MIMO detector 710 receives a plurality of signalstreams received though the antennas 705, determines a reduced initialcandidate set, calculates EDs, generates an LLR for x₀, and provides thegenerated LLR for x₀ to the decoder 715.

Further, when the MIMO detector 710 determines the reduced initialcandidate set using prior information, e.g., using the MMSE-SIC schemeusing the prior information and the approximation over MMSE-SIC schemeusing the prior information, the decoder 715 may feedback priorinformation to the MIMO detector 710.

As described above, certain embodiments of the present disclosureprovide low complexity schemes for soft detection of high order QAMsymbols in MIMO channels, with or without prior information (from adecoder).

Further, ICR schemes are provided, which reduce the number of EDcalculations by half in comparison with soft ML detection, and as aresult, may reduce the total hardware size by almost 50%.

Additionally, for initial candidate set selection, only the I and Qsigns of the target layer need to be detected, which simplifies MMSEdetection.

Further, to improve the accuracy of the initial candidate set, priorinformation is used in initial candidate set selection with MMSE-SIC. Toavoid computational complexity of MMSE-SIC, a low complexity initialdetection with prior information is also provided.

Depending on the embodiment of the present disclosure, steps and/oroperations in accordance with the present disclosure may occur in adifferent order, or in parallel, or concurrently for different epochs,etc., in different embodiments, as would be understood by one ofordinary skill in the art.

Depending on the embodiment, some or all of the steps and/or operationsmay be implemented or otherwise performed, at least in part, on aportable device. “Portable device” as used herein refers to anyportable, mobile, or movable electronic device having the capability ofreceiving wireless signals, including, but not limited to, multimediaplayers, communication devices, computing devices, navigating devices,etc. Thus, mobile devices include, but are not limited to, laptops,tablet computers, Portable Digital Assistants (PDAs), mp3 players,handheld PCs, Instant Messaging Devices (IMD), cellular telephones,Global Navigational Satellite System (GNSS) receivers, watches, camerasor any such device which can be worn and/or carried on one's person orremain in close proximity to the person.

Depending on the embodiment, some or all of the steps and/or operationsmay be implemented or otherwise performed, at least in part, using oneor more processors running instruction(s), program(s), interactive datastructure(s), client and/or server components, where suchinstruction(s), program(s), interactive data structure(s), client and/orserver components are stored in one or more non-transitorycomputer-readable media. The one or more non-transitorycomputer-readable media may be instantiated in software, firmware (orembedded software), hardware, and/or any combination thereof. Moreover,the functionality of any “module” discussed herein may be implemented insoftware, firmware, hardware, and/or any combination thereof.

The one or more non-transitory computer-readable media and/or means forimplementing/performing one or more operations/steps/modules ofembodiments of the present disclosure may include, without limitation,application-specific integrated circuits (ASICs), standard integratedcircuits, controllers executing appropriate instructions (includingmicrocontrollers and/or embedded controllers), field-programmable gatearrays (FPGAs), complex programmable logic devices (CPLDs), and thelike. Some or all of any system components and/or data structures mayalso be stored as contents (e.g., as executable or other non-transitorymachine-readable software instructions or structured data) on anon-transitory computer-readable medium (e.g., as a hard disk; a memory;a computer network or cellular wireless network or other datatransmission medium; or a portable media article to be read by anappropriate drive or via an appropriate connection, such as a DVD orflash memory device) so as to enable or configure the computer-readablemedium and/or one or more associated computing systems or devices toexecute or otherwise use or provide the contents to perform at leastsome of the described techniques. Some or all of any system componentsand data structures may also be stored as data signals on a variety ofnon-transitory computer-readable transmission mediums, from which theyare read and then transmitted, including across wireless-based andwired/cable-based mediums, and may take a variety of forms (e.g., aspart of a single or multiplexed analog signal, or as multiple discretedigital packets or frames). Such computer program products may also takeother forms in other embodiments. Accordingly, embodiments of thisdisclosure may be practiced in any computer system configuration.

Thus, the term “non-transitory computer-readable medium” as used hereinrefers to any medium that includes the actual performance of anoperation (such as hardware circuits), that includes programs and/orhigher-level instructions to be provided to one or more processors forperformance/implementation (such as instructions stored in anon-transitory memory), and/or that includes machine-level instructionsstored in, e.g., firmware or non-volatile memory. Non-transitorycomputer-readable media may take many forms, such as non-volatile andvolatile media, including but not limited to, a floppy disk, flexibledisk, hard disk, RAM, PROM, EPROM, FLASH-EPROM, EEPROM, any memory chipor cartridge, any magnetic tape, or any other magnetic medium from whicha computer instruction can be read; a CD-ROM, DVD, or any other opticalmedium from which a computer instruction can be read, or any othernon-transitory medium from which a computer instruction can be read.

While certain embodiments of the present disclosure have been shown anddescribed herein, it will be understood by those of ordinary skill inthe art that various changes in form and details may be made thereinwithout departing from the spirit and scope of the present disclosure,i.e., the invention is not limited to any embodiments described herein,but is defined by the appended claims and their equivalents.

What is claimed is:
 1. A method comprising: receiving a plurality ofsignals including Q-order quadrature amplitude modulation (QAM) symbols;determining a reduced candidate set including C potential candidates,where C is less than Q; calculating Euclidean distances (EDs) based onthe reduced candidate set; and generating log-likelihood ratio (LLR)information based on the calculated EDs.
 2. The method of claim 1,further comprising providing the generated LLR information to a decoder.3. The method of claim 1, wherein Q is 256 and C is
 128. 4. The methodof claim 1, wherein determining the reduced candidate set comprises:determining a linear minimum mean square error (MMSE); and selecting thereduced candidate set based on the determined linear MMSE.
 5. The methodof claim 4, wherein the linear MMSE is determined using: $\begin{matrix}{{\hat{x}}_{0} = {\frac{1}{{ch}_{pow}}\left( {{\left( {{h_{1}}^{2} + \sigma^{2}} \right)h_{0}^{H}y} - {h_{0}^{H}h_{1}h_{1}^{H}y}} \right)}} \\{= \frac{{\left( {{\left( {{h_{1}}^{2} + \sigma^{2}} \right){h_{0}}^{2}} - {{h_{0}^{H}h_{1}}}^{2}} \right)x_{0}} + {\sigma^{2}{{h_{0}^{H}h_{1}}}^{2}x_{1}} + n^{\prime}}{{ch}_{pow}}}\end{matrix}$ where {circumflex over (x)}₀ represents an initial MMSEestimation, y=[y₀, . . . , y_(r−1)]^(T) is an n_(R)×1 receive signalvector, x=[x₀, x₁]^(T) is a 2×1 transmit signal vector, H=[h₀, h₁] is ann_(R)×2 channel coefficient matrix, h_(i)=[h_(i,0), . . . , h_(i,n) _(R)⁻¹]^(T), h_(i,j) represents a channel between i-th transmit and j-threceive antennas, n is an additive white Gaussian noise vector withcovariance E{nn^(H)}=σ²I, ch_(pow)=(|h₁|²+σ²)(|h₀|²+σ²)−|h₀ ^(H)h₁|²,and n′=(|h₁|²+σ²)h₀ ^(H) n−h₀ ^(H)h₁h₁ ^(H)n.
 6. The method of claim 5,wherein the LLR information is generated using:${L_{A}\left( b_{0,l} \right)} \approx {{\log \frac{{\max\limits_{{{{x_{0} \in {X_{128}{({\hat{x}}_{0})}}}\&}b_{0,l}} = 0}{\max\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}{P\left( b_{m,n} \right)}}}}}\;}{{\max\limits_{{{{x_{0} \in {X_{128}{({\hat{x}}_{0})}}}\&}b_{0,l}} = 1}{\max\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}{P\left( b_{m,n} \right)}}}}}\;}} + {L_{a}\left( b_{0,l} \right)}}$where b_(0,l) represents an 1-th bit of symbol x₀, L_(A)(b_(0,l)) is aposteriori LLR of b_(0,l) and X₁₂₈({circumflex over (x)}₀) is a set of128 initial candidates selected using the initial MMSE estimation{circumflex over (x)}₀.
 7. The method of claim 1, wherein determiningthe reduced candidate set comprises: receiving prior information;determining a linear minimum mean square error (MMSE) soft interferencecancellation (MMSE-SIC); and selecting the reduced candidate set basedon the determined linear MMSE-SIC.
 8. The method of claim 7, wherein theprior information includes at least one of self-prior information andcross-prior information.
 9. The method of claim 7, wherein the linearMMSE-SIC is determined using: $\begin{matrix}{{\hat{x}}_{0} = {\frac{1}{{ch}_{pow}}\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}h_{0}^{H}y} - {\lambda_{0}^{2}\lambda_{1}^{2}h_{0}^{H}h_{1}h_{1}^{H}y} - {\sigma^{2}\lambda_{0}^{2}h_{0}^{H}h_{1}\mu_{1}} -} \right.}} \\{\left. {\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}{h_{0}}^{2}} - {\lambda_{0}^{2}\lambda_{1}^{2}{{h_{0}^{H}h_{1}}}^{2}}} \right)\mu_{0}} \right) + \mu_{0}} \\{= {\frac{1}{{ch}_{pow}}\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}h_{0}^{H}y} - {\lambda_{0}^{2}\lambda_{1}^{2}h_{0}^{H}h_{1}h_{1}^{H}y} - {\sigma^{2}\lambda_{0}^{2}h_{0}^{H}h_{1}\mu_{1}} +} \right.}} \\\left. {{\sigma^{2}\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)}\mu_{0}} \right)\end{matrix}$ where y=[y₀, . . . , y_(r−1)]^(T) is an n_(R)×1 receivesignal vector, x=[x₀, x₁]^(T) is a 2×1 transmit signal vector, H=[h₀,h₁] is an n_(R)×2 channel coefficient matrix, h_(i)=[h_(i,0), . . . ,h_(i,n) _(R) ⁻¹]^(T), h_(i,j) represents a channel between i-th transmitand j-th receive antennas, ch_(pow)=(|h₁|²+σ²)(|h₀|²+σ²)−|h₀ ^(H)h₁|²,μ_(i)=E{x_(i)}, and ${E\left\{ {\begin{bmatrix}{x_{0} - \mu_{0}} \\{x_{1} - \mu_{1}}\end{bmatrix}\begin{bmatrix}{x_{0}^{*} - \mu_{0}^{*}} & {x_{1}^{*} - \mu_{1}^{*}}\end{bmatrix}} \right\}} = {\begin{bmatrix}\lambda_{0}^{2} & 0 \\0 & \lambda_{1}^{2}\end{bmatrix}.}$
 10. The method of claim 1, wherein determining thereduced candidate set comprises: receiving prior information; estimatinga minimum mean square error (MMSE) without using the prior information;slicing over the estimated MMSE based on the received prior information;and selecting the reduced candidate set based on the sliced estimatedMMSE.
 11. An apparatus comprising: a plurality of antennas; and amultiple input multiple output (MIMO) detector that receives, via theplurality of antennas, a plurality of signals including Q-orderquadrature amplitude modulation (QAM) symbols, determines a reducedcandidate set including C potential candidates, where C is less than Q,calculates Euclidean distances (EDs) based on the reduced candidate set,and generates log-likelihood ratio (LLR) information based on thecalculated EDs.
 12. The apparatus of claim 11, wherein the MIMO detectorprovides the generated LLR information to a decoder.
 13. The apparatusof claim 11, wherein Q is 256 and C is
 128. 14. The apparatus of claim11, wherein the MIMO detector determines the reduced candidate set by:determining a linear minimum mean square error (MMSE); and selecting thereduced candidate set based on the determined linear MMSE.
 15. Theapparatus of claim 14, wherein the MIMO detector determines the linearMMSE using: $\begin{matrix}{{\hat{x}}_{0} = {\frac{1}{{ch}_{pow}}\left( {{\left( {{h_{1}}^{2} + \sigma^{2}} \right)h_{0}^{H}y} - {h_{0}^{H}h_{1}h_{1}^{H}y}} \right)}} \\{= \frac{{\left( {{\left( {{h_{1}}^{2} + \sigma^{2}} \right){h_{0}}^{2}} - {{h_{0}^{H}h_{1}}}^{2}} \right)x_{0}} + {\sigma^{2}{{h_{0}^{H}h_{1}}}^{2}x_{1}} + n^{\prime}}{{ch}_{pow}}}\end{matrix}$ where {circumflex over (x)}₀ represents an initial MMSEestimation, y=[y₀, . . . , y_(r−1)]^(T) is an n_(R)×1 receive signalvector, x=[x₀, x₁]^(T) is a 2×1 transmit signal vector, H=[h₀, h₁] is ann_(R)×2 channel coefficient matrix, h_(i)=[h_(i,0), . . . , h_(i,n) _(R)⁻¹]^(T), h_(i,j) represents a channel between i-th transmit and j-threceive antennas, n is an additive white Gaussian noise vector withcovariance E{nn^(H)}=σ²I, ch_(pow)=(|h₁|²+σ²)(|h₀|²+σ²)−|h₀ ^(H)h₁|²,and n′=(|h₁|²+σ²)h₀ ^(H) n−h₀ ^(H)h₁h₁ ^(H)n.
 16. The apparatus of claim15, wherein the MIMO detector generates the LLR information using:${L_{A}\left( b_{0,l} \right)} \approx {{\log \frac{{\max\limits_{{{{x_{0} \in {X_{128}{({\hat{x}}_{0})}}}\&}b_{0,l}} = 0}{\max\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}{P\left( b_{m,n} \right)}}}}}\;}{{\max\limits_{{{{x_{0} \in {X_{128}{({\hat{x}}_{0})}}}\&}b_{0,l}} = 1}{\max\limits_{x_{1}}{^{- \frac{{{y - {Hx}}}^{2}}{\sigma^{2}}}{\prod\limits_{{({m,n})} \neq {({i,l})}}{P\left( b_{m,n} \right)}}}}}\;}} + {L_{a}\left( b_{0,l} \right)}}$where b_(0,l) represents the 1-th bit of symbol x₀, L_(A)(b_(0,l)) is aposteriori LLR of b_(0,l) and X₁₂₈({circumflex over (x)}₀) is the set of128 initial candidates selected using the initial MMSE estimation{circumflex over (x)}₀.
 17. The apparatus of claim 11, wherein the MIMOdetector determines the reduced candidate set by: receiving priorinformation; determining a linear minimum mean square error (MMSE) softinterference cancellation (MMSE-SIC); and selecting the reducedcandidate set based on the determined linear MMSE-SIC.
 18. The apparatusof claim 17, wherein the prior information includes at least one ofself-prior information and cross-prior information.
 19. The apparatus ofclaim 17, wherein the MIMO detector determines the linear MMSE-SICusing: $\begin{matrix}{{\hat{x}}_{0} = {\frac{1}{{ch}_{pow}}\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}h_{0}^{H}y} - {\lambda_{0}^{2}\lambda_{1}^{2}h_{0}^{H}h_{1}h_{1}^{H}y} - {\sigma^{2}\lambda_{0}^{2}h_{0}^{H}h_{1}\mu_{1}} -} \right.}} \\{\left. {\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}{h_{0}}^{2}} - {\lambda_{0}^{2}\lambda_{1}^{2}{{h_{0}^{H}h_{1}}}^{2}}} \right)\mu_{0}} \right) + \mu_{0}} \\{= {\frac{1}{{ch}_{pow}}\left( {{\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)\lambda_{0}^{2}h_{0}^{H}y} - {\lambda_{0}^{2}\lambda_{1}^{2}h_{0}^{H}h_{1}h_{1}^{H}y} - {\sigma^{2}\lambda_{0}^{2}h_{0}^{H}h_{1}\mu_{1}} +} \right.}} \\\left. {{\sigma^{2}\left( {{\lambda_{1}^{2}{h_{1}}^{2}} + \sigma^{2}} \right)}\mu_{0}} \right)\end{matrix}$ where y=[y₀, . . . , y_(r−1)]^(T) is an n_(R)×1 receivesignal vector, x=[x₀, x₁]^(T) is a 2×1 transmit signal vector, H=[h₀,h₁] is an n_(R)×2 channel coefficient matrix, h_(i)=[h_(i,0), . . . ,h_(i,n) _(R) ⁻¹]^(T), h_(i,j) represents a channel between i-th transmitand j-th receive antennas, ch_(pow)=(|h₁|²+σ²)(|h₀|²+σ²)−|h₀ ^(H)h₁|²,μ_(i)=E{x_(i)}, and ${E\left\{ {\begin{bmatrix}{x_{0} - \mu_{0}} \\{x_{1} - \mu_{1}}\end{bmatrix}\begin{bmatrix}{x_{0}^{*} - \mu_{0}^{*}} & {x_{1}^{*} - \mu_{1}^{*}}\end{bmatrix}} \right\}} = {\begin{bmatrix}\lambda_{0}^{2} & 0 \\0 & \lambda_{1}^{2}\end{bmatrix}.}$
 20. The apparatus of claim 11, wherein the MIMOdetector determines the reduced candidate set by: receiving priorinformation; estimating a minimum mean square error (MMSE) without usingthe prior information; slicing over the estimated MMSE based on thereceived prior information; and selecting the reduced candidate setbased on the sliced estimated MMSE.
 21. A system on chip comprising: aMIMO detector that receives a plurality of signals including Q-order QAMsymbols, determines a reduced candidate set including C potentialcandidates, where C is less than Q, calculates Euclidean distances (EDs)based on the reduced candidate set, and generates log-likelihood ratio(LLR) information based on the calculated EDs; and a decoder thatdecodes the signals using the LLR information.